Write equations of forces in terms of the relative acceleration for the blocks shown in and find the relation between m and M so that system remains at rest

Explanation

Total length of the cord
`y_(1)+3y_(2)+l_(0)=l`
here `y_(1)` and `y_(2)` changes `l_(0)` and l remain constant
Differetiating Eq. (i) with respect to time, we get
`dy_(1)/(dt)+3(dy_(2))/(dt)=0`
or `v_(1)+3v_(2)=0` ....(ii)
also `a_(1)+3a_(2)=0`
which gives `a_(1)=3a_(2)` (magnitude)
if we take `a_(2)=a` up

then `a_(1)=3a` down
By Newton's second law we have
`mg-T=m(3a)`...(iii)
and `3T-Mg=Ma` ....(iv)
Solving Eqs (iii) and (iv)
`a=((3m-M))/((9m+M))g`
if system remains at rest `a=0`
`:. 3m=M`